Normalised lines

When we looked at L(x) = ax+b we saw that we could not make a vertical line with that. A vertical line had the equation x=c with c being some number.

Mathematicians hate exceptions. For the function of a line, they prefer to write the normalised form:

ax+by+c=0

How is this different? We now see three numbers: a, b and c whereas before we had only two (a and b).

But in fact there are only two that you can choose, the third is "dependent".

Which two? Well, we are not allowed ever to divide by zero, but suppose a is not equal to zero. Then we are allowed to divide both sides of the equation by a:

x+(b/a)y+(c/a) = 0

and now there are only two numbers: b/a and c/a (after you have worked them out). If a is zero but b is not zero then we can divide by b and get:

(a/b)x+y+(c/b) = 0

and again we have only two numbers. Likewise if c is not zero we get

(a/c)x+(b/c)y+1 = 0

But note one very important thing about using the form

ax+by+c=0

instead of the form y = ax+b : in the normalised form any of the three numbers can be zero.

Cases of the normalised form

And now we have three cases:

a=0
then we get by+c=0 or, if we subtract c from both sides and divide by b:y=(-c/b)
or, y is a constant, the horizontal line.

b=0
then we get ax+c=0 and if we do the same as in 1 we get:x=(-c/a)
or, x is a constant, the vertical line!

c=0
then we get ax+by=0 and if we subtract ax from both sides we getby=-ax
and then divide by b:y=(-a/b)x
which is just a line through the origin (y=0 if we make x=0).

So! The normalised form lets us get all possible lines, including the vertical ones, without having to make an exception.

Lo and behold, Grapher can do it too:

Problems / Tasks

1. Play with Grapher to see what influences a, b and c have if you use the normal form.

2. Maths does not play with numbers. Working with numbers is called reckoning or arithmetic, it is not mathematics really. Looking at ax+by+c=0 and not using any numbers, only the letters a, b, c, can you write out what the slope of the line is? Remember that the slope is the amount by which y goes up (or down if the slope is negative) when x changes by one unit.

3. As in problem 2, write out where the line crosses the x-axis and where it crosses the y-axis. What do you observe in the answer?